# Path Integrals and Brownian Motion with non-standard kinetic term

Recently I've been thinking about $$1d$$ Path Integrals of some theories with non-standard Lagrangians. The adjective non-standard meaning that the Lagrangian $$\mathcal{L} \neq \frac{1}{2} m \dot{x}^2 - V(x)$$, and has a more general form $$\mathcal{L}= \mathcal{L(x, \dot{x}, \ddot{x}, \dots)}$$.

In my understanding of the Path Integral mathematical formalization of Quantum Mechanics we usually identify the one dimensional functional integral (in Euclidean signature) $$\int \mathcal{D}[X] e^{- \int \mathrm{d} \tau \frac{1}{2} m \,\dot{x}^2 + V(x) },$$ with the following integral: $$\int e^{-\int \mathrm{d}\tau V(x)} \mathrm{d}W_{t}$$ where $$\mathrm{d}W_{t}$$ denotes the mathematically well-defined Weiner measure. As far as I understand, this identification emerges in a natural way from the physical picture by discretizing the "time range" of the functional integral (and crucially stems from the presence of the kinetic term) and allows to make contact with Brownian Motions and to the very physically appealing picture of "sum over all paths": $$\int \mathcal{D}[X] e^{- \int \mathrm{d} \tau \frac{1}{2} m \,\dot{x}^2 + V(x) } = \sum_{\text{Brownian paths}} e^{-S}.$$

My question is: Is this picture valid for theories with a Lagrangian that does not have a manifest kinetic term of the form $$\frac{1}{2} m \dot{x}^2?$$ And more in general, is there a geometric interpretation for such theories?

I have not worked with path integrals in the context of QM, but the result I am aware of in the context of stochastic process is that you get the Onsager-Machlup Lagrangian, $$L(x,\dot{x})=\frac{1}{2D} |\dot{x}-f(x)|^2+\alpha\partial_x f(x),$$ when you do the path-integral representation of the stochastic differential equation $$dx=f(x)dt+\sqrt{D} dW_t,$$ where $$\alpha$$ parametrizes the noise discretization ($$\alpha=0$$ in Itô interpretation).
This Lagrangian has a kinetic term proportional to $$\dot{x}^2$$, but it is "non-standard" according to your definition because it has a term proportional to $$x\dot{x}$$.
If you check the derivation of the Onsager-Machlup Lagrangian, you can see that the kinetic term comes from the fact the SDE is of first order. Maybe Lagrangians with $$\ddot{x}$$ terms can be derived from second-order SDEs (but I am not sure and I don't know references about this particular topic).